A functional estimate for lower bound of convolution function with concave power law functions can be obtain by the following lower bound estimate, that will become essential later on the calculation of propagation of \(L^p\) -Banach norms for any \(1<p\le \infty \) . This is a fundamental result related to a lower control for the convolution operator associated to the collision frequency corresponding to the negative contributions of the moments associated to the collision operator \(Q(f,f)(v)\) . This a new result, developed for probability densities and independent of being solution of the Boltzmann equation. A first notion of this lower bound was introduced by the authors for the analysis of stability and error estimates for numerical approximations to the Boltzmann equation by conservative spectral schemes as shown in [1]}, and it is reproduced here in a more analytical framework. Modification of this results were employed in to obtain existence results and qualitative results for the one dimensional dissipative Boltzmann type equation for polymers [2]}, and more recently used in the well-posedness of of solution of a gas mixture modeled by a system of Boltzmann equation with disparate masses [3]} and for polyatomic gas taking into account the exchange of internal energies [4]}

